The Chaotic Universe

The following sections are included:

• FOREWORD

• CONTENTS

The following sections are included:

• Introduction

• The determinism in nature

• The problem of time in physics and in philosophy, towards reconciliation.

• Reversibility and irreversibility in nature

• The irreversibility and the basic physics laws

• Irreversibility, bifurcations and history

• Science as history of nature

• The irreversibility in the example of deterministic Chaos

• Irreversibility and thermodynamical systems

• The mathematics of time

• The extension of classical mechanics or quantum mechanics, example in field theory and in elementary particle physics

• Conclusions

I will discuss in this review the cosmic gravitational clustering on scales equal to and larger than galaxy clusters. The emphasis here is on the dynamical features of the structure formation of clusters and beyond, and their constraints on cosmological parameters.

The properties of the Cosmic Microwave Background (CMB) radiation must be different in flat, positively and negatively curved universes. This fact leads to a direct way of determining the geometry of the universe. The signature of the predicted effect of geodesic mixing, i.e., of the ‘chaotic’ behavior of photon beams in negatively curved spaces peculiar to Anosov systems with strong statistical properties, has been detected while studying the COBE-DMR 4-year data. This possible observation of the negative curvature of the universe suggests the need to search for more effective ways to analyze the CMB data expected from forthcoming high precision experiments. Information theory offers such a descriptor for the CMB sky maps — the Kolmogorov complexity — as well as provides novel insight into the problem of the loss of information and time asymmetry in a hyperbolic universe.

Galactic globular clusters, which are ancient building blocks of our Galaxy, represent a very interesting family of stellar systems in which some fundamental dynamical processes have taken place on time scales shorter than the age of the universe. In contrast with galaxies, these star clusters represent unique laboratories for learning about two-body relaxation, mass segregation from equipartition of energy, stellar collisions, stellar mergers, core collapse, and tidal disruption. This review briefly summarizes some of the tremendous developments that have taken place during the last two decades. It ends with some recent results on tidal tails around galactic globular clusters and on a very massive globular cluster in M31.

We have shown before that chaotic stellar motion should be present in galactic satellites. Here we extend our study to satellites modelled according to the Heggie and Ramamani recipe which has the advantage of providing a distribution function. We are thus able to determine the fraction of chaotic orbits, which turns out to be very high (23% – 65%, depending on the model). Most of the regular orbits are short-axis tubes, while box orbits make up less than 10% of the total. A tentative scenario is described to explain the relationships between chaoticity and orbital energy.

Frequency Map Analysis is a numerical method based on Fourier techniques which provides a clear representation of the global dynamics of many multi-dimensional systems, and which is particularly useful for systems of 3 degrees of freedom and more. It has been applied to a large variety of dynamical systems, starting with solar system dynamics. Here are presented the theoretical foundation of the method, and a brief account of its application in galactic dynamics.

The topic of this conference is “The Chaotic Universe”. One of the main achievements of last century has been to relate chaos in fluids to their thermodynamics. It is our purpose to make connection between chaos in gravitation and standard thermodynamics. Though there have been many previous steps and attempts, so far no convincing conclusion has been reached. After explaining how the approach works for glasses, we shall discuss the thermodynamics of two specific systems: black holes and globular star clusters. In both cases we point out that the dynamics satisfies the first and second law of thermodynamics, though negative specific heats occur.

The interstellar medium is structured as a hierachy of gas clouds, that looks selfsimilar over 6 orders of magnitude in scales and 9 in masses. This is one of the more extended fractal in the Universe. At even larger scales, the ensemble of galaxies looks also self-similar over a certain ranges of scales, but more limited, may be over 3-4 orders of magnitude in scales. These two fractals appear to be characterized by similar Hausdorff dimensions, between 1.6 and 2. The various interpretations of these structures are discussed, in particular formation theories based on turbulence and self-gravity. In the latter, the fractal ensembles are considered in a critical state, as in second order phase transitions, when large density fluctuations are observed, that also obey scaling laws, and look self-similar over an extended range.

Recent development of a functional analysis in a wider class of function spaces than the Hilbert space shows that irreversibility is a rigorous dynamical phenomenon. No anthropomorphic principle is necessary to explain irreversibility. Beyond the Hilbert space the Liouville-van Neumann operator admits irreducible complex spectral representations. Here, “irreducible” means that the description cannot be implemented to wave functions, and “complex” means that time-symmetry is broken. Decoherence for quantum Brownian motion is discussed as an example of irreversible processes. An essential element in understanding of the quantum decoherence is the “extensivity” of the quantities that characterise thermodynamic systems. It is not necessary that surrounding particles are in a thermal equilibrium as even pure states may dynamically evolve toward mixed states through the decoherence process. Lower bound of decoherence time is given by the quantum Zeno time.

I compare the mean power spectrum of galaxies with theoretical models and discuss possibilities to explain the observed power spectrum. My principal conclusion is that some of the presently accepted cosmological paradigms need revision if the available observational data represent a fair sample of the Universe.

We illustrate what we call the Boltzmann-Jeans effect, namely that the relaxation time to equilibrium for oscillators of a given frequency in general increases exponentially fast with the frequency. This leads one to expect that the blackbody spectrum (divided by the density of modes) might present a plateau in the low frequency region.

We perform a discrete wavelet analysis of the COBE-DMR 4yr sky maps and find a significant scale-scale correlation on angular scales from about 11 to 22 degrees in the DMR face centered on the North Galactic Pole. This non-Gaussian signature does not arise either from the known foregrounds or the correlated noise maps, nor is it consistent with upper limits on the residual systematic errors in the DMR maps. This correlation may be evident the long-time tail correlation of the primeval density perturbations.

The relation between the thermodynamical and cosmological arrows of time is usually viewed in the context of the initial conditions of the Universe. It is a necessary but not sufficient condition for ensuring the thermodynamical arrow. We point out that in the Friedmann-Robertson-Walker Universe with negative curvature k = −1 there is the second necessary ingredient. It is based on the geodesic mixing - the dynamical instability of motion along null geodesics in hyperbolic space. Kolmogorov (algorithmic ) complexity as a universal and experimentally measurable concept can be very useful in description of this chaotic behavior using the data on Cosmic Microwave Background radiation. The formulated curvature anthropic principle states the negative curvature as a necessary condition for the time asymmetric Universe with an observer.

We review recent progress in understanding the role of chaos in influencing the structure and evolution of galaxies. The orbits of stars in galaxies are generically chaotic: the chaotic behavior arises in part from the intrinsically grainy nature of a potential that is composed of point masses. Even if the potential is assumed to be smooth, however, much of the phase space of non-axisymmetric galaxies is chaotic due to the presence of central density cusps or black holes. The chaotic nature of orbits implies that perturbations will grow exponentially and this in turn is expected to result in a diffusion in phase space. We show that the degree of orbital evolution is not well predicted by the growth rate of infinitesimal perturbations, i.e. by the Liapunov exponent. A more useful criterion is whether perturbations continue to grow exponentially until their scale is of order the size of the system. We illustrate these ideas in a potential consisting of N fixed point masses. Liapunov exponents are large for all values of N, but orbits become increasingly regular in their behavior as N increases; the reason is that the exponential divergence saturates at smaller and smaller distances as N is increased. The objects which lend phase space its structure and impede diffusion are the invariant tori; in the triaxial potentials we consider, a large fraction of the tori correspond to resonant (thin) orbits and their associated families of regular orbits. Perturbations to the potential destroy the resonant tori. When only a few stable resonances remain, we find that the phase space distribution of an ensemble of chaotic orbits evolves rapidly toward a nearly stationary state. This mixing process is shown to occur on timescales of a few crossing times in triaxial potentials containing massive central singularities, consistent with the rapid evolution observed in N-body simulations of galaxies with central black holes.

We analyse different tools to study global and local dynamics in Hamiltonian Systems focusing the discussion on Galactic Dynamics. In this direction we introduce the so-called conditional entropy of nearby orbits as an alternative technique. The numerical evidence at hand reveals that it appears to be very useful to understand the global and local structure of the phase space as well as to derive a good estimation of the largest Lyapunov characteristic number but in realistic physical times. The required computational effort is relatively small, almost the same needed to compute the latter number but over shorter motion times. Comparisons with other well known techniques, like the frequency map analysis, are presented. At least for the examples considered here, the proposed tool provides detailed information about the dynamics, even more than any other technique used before in this kind of studies.

The relation between relaxation, the time scale of Lyapunov instabilities, and the Kolmogorov-Sinai time in a one-dimensional gravitating sheet system is studied. Both the maximum Lyapunov exponent and the Kolmogorov-Sinai entropy decrease as proportional to N^−1/5. The time scales determined by these quantities evidently differ from any type of relaxation time found in the previous investigations. The relaxation time to quasiequilibria (microscopic relaxation) is found to have the same N-dependence as the inverse of the minimum positive Lyapunov exponent. The relaxation time to the final thermal equilibrium differs to the inverse of the Lyapunov exponents and the Kolmogorov-Sinai time.

Symplectic algorithms (SA’s) are known as the best suited ones for a careful numerical integration of the dynamics of Hamiltonian Systems on very long runs. This is particularly relevant when there are no obvious additional conserved quantities other than the energy to be used to check the reliability of the numerical results. Chaotic systems belong clearly to this class. Moreover, SA’s can be naturally extended to study the evolution of perturbations to the dynamical trajectories of Hamiltonian Systems. Due to their peculiarities, Dynamical Systems (DS’s) originating from General Relativity (GR) require very accurate numerical integrations, as, for them, even the qualitative features of the dynamics depend very sensitively on the exact conservation of the (vanishing) value of the energy, so that it is essential to devise a scheme able to preserve the Hamiltonian structure on very long runs. Besides, GR cosmological models demand to the numerical algorithms even more, because, at variance with usual Hamiltonians, their dynamics is characterized by a secular evolution which prevents fixed step-size schemes to work effectively.

We illustrate here some topical points related to the issues above, applying to a DS of cosmological interest a recently proposed general procedure to extend SA’s to use adaptive time-steps.

The Ricci curvature criterion is used for the investigation of the relative instability of several configurations of N-body gravitating systems. It is shown, that the systems with double massive centers are more unstable than the homogeneous systems and those with one massive center. In general this shows the efficiency of the Ricci curvature method introduced by Gurzadyan and Kocharyan (1987) for the study of N-body systems via relatively simple calculations, i.e. for small N, and hence small computer resources.

A new direct N-body code, NBODY6, supersedes the similar code NBODY5 which has been in general use since the early 1980s. This code incorporates several recent developments which result in improved accuracy as well as performance. It is suitable for studying a variety of problems involving point-mass interactions, with emphasis on realistic star cluster models.

We know that an adequate model plays the important role in physics and astrophysics. We consider samples of gravitational lens models when numerical simulations or a degeneracy of the gravitational lens model may yield non-adequate properties of the physical problems.

An accurate estimate of the baryon fraction f_b in galaxy clusters plays a potentially important role in the ‘direct’ measurement of the cosmic density parameter, Ω_m. In this talk, we first present the X-ray measurements of f_b among > 200 clusters under the conventional assumption that the intracluster gas is in isothermal and hydrostatic equilibrium with the underlying gravitational potential of clusters. Using a subsample of 80 clusters, we further examine the dependence of the baryon fraction upon the X-ray temperature and cluster radius, and compare the theoretically expected and observationally fitted X-ray luminosity (L,_x)-temperature (T) relation. While both the f_b-T and L_x,-T relations are consistent with the prediction of standard models that galaxy clusters of different temperatures share a common baryon fraction, the baryon fraction is a monotonically increasing function of cluster radius in the framework of isothermal β model and hydrostatic equilibrium for intracluster gas, and therefore its asymptotic trend at large radius cannot reach a universal value unless a mildly increasing (for 1/3 < β < 2/3) or decreasing (for β > 2/3) temperature profile with radius is invoked. Consequently, our current estimate of Ω_m based on f_b within a certain cutoff radius could contain large uncertainties.

For investigation of evolution of orbits in gravitating systems the probabilities of given variations of orbit integrals are found for the case of binary interactions in plane and 3D systems. The probabilities are the kernels of kinetic equation. The method takes into account not only small but also large variations of orbit integrals (close encounters). The evolution of distribution of orbit integrals, including the changing of degree of the radial orbit stretch, in spherical cluster is investigated.

We investigate the simplest cosmological model with the massive real scalar noninteracting inflaton field minimally coupled to gravity. The classification of trajectories in closed minisuperspace Friedmann-Robertson-Walker model is presented. The possible fractal nature of a set of infinitely bounced trajectories is discussed. The chaotical dynamics is studied in different Friedmann-Robertson- Walker cosmological models with scalar (inflaton) field and hydronamical matter. The topological entropy is calculated for some particular cases. Suggested scheme can be easily generalized for wide class of models. Different methods of calculation of topological entropy are compared.

All inhabitants of this universe, from galaxies to people, are finite. Yet the universe itself is often assumed to be infinite. If instead the universe is topologically finite, then light and matter can take chaotic paths around the compact geometry. Chaos may lead to ordered features in the distribution of matter throughout space.

I review the main ideas of the pre-big-bang cosmology scenario emphasizing the role of different boundary conditions in comparison to the standard ones which appear in quantum cosmology. My main issue is duality symmetry - a very general feature of string theory - and its role in suppressing chaos in Bianchi type IX “Mixmaster” universes within the framework of the tree-level low-energy-effectiveactions for strings. Finally, I discuss the ways to possibly ‘generate’ chaos in string cosmology by admitting dilaton potential/massive string modes, more spacetime dimensions or nonlinear Yang-Mills-Lorentz-Chern-Simons terms into the action.

The talk presents a short review of general character on the status of the chaotic phenomena in Cosmology

Here we present a detailed analysis of the asymptotic dynamical behaviour characterizing, close enough to the cosmological singularity, the Bianchi type VIII and IX cosmological models (the so-called mixmaster universe). After a brief review of the Belinski-Khalatnikov-Lifshitz (BKL) piecewise iterative representation of the mixmaster evolution, we search for a statistical reformulation of the dynamical problem in terms of continuous variables. On the base of a standard Arnowitt- Deser-Misner (ADM) Hamiltonian approach, relative to suitable Misner-Chitre-like variables, we derive the asymptotic statistical distribution for the system dynamics, which admits a stationary uniform limit.

The dynamics of closed scalar field FRW cosmological models is studied for several types of exponentially and more than exponentially steep potentials. The parameters of scalar field potentials which allow a chaotic behaviour are found from numerical investigations. It is argued that analytical studies of equation of motion at the Euclidean boundary can provide an important information about the properties of chaotic dynamics. Several types of transition from chaotic to regular dynamics are described.

Relativistic cluster may exist within galactic centers and mimic the presence of massive black holes. The main evolution time-scale of these clusters are investigated with particular emphasis to the gravitational wave damping time. The detection of gravitational waves is also explored by using the capabilities of the VIRGO experiment.

I review recent results derived from numerical simulations of the turbulent interstellar medium (ISM), in particular concerning the nature and formation of turbulent clouds and consequences thereof, methods for comparing the structure in simulations and observations, and the effects of projection of three-dimensional structures onto two dimensions. Clouds formed as turbulent density fluctuations are probably not confined by thermal pressure, but rather their morphology may be determined by the large-scale velocity field. Also, they may have shorter lifetimes than normally believed, as the large-scale turbulent modes have larger associated velocities than the clouds’ internal velocity dispersions. Structural characterization algorithms have started to distinguish the best fitting simulations to a particular observation, and have posed several new questions, such as the nature of the observed line width-size relation and of the relation between the structures seen in channel maps and the true spatial distribution of the density and velocity fields. The velocity field apparently dominates the morphology seen in intensity channel maps, at least in cases when the density field exhibits power spectra steep enough. Furthermore, channel maps may contain a spurious excess of small-scale structure compared to the original 3D density and velocity fields.

The complexity of observed molecular cloud structures prevents any simple description and complicates the comparison of observations with cloud models. We provide a short overview on the techniques that are applied to parameterize the cloud structure. Several independent parameters have to be combined including measures for the isotropic density or intensity scaling behaviour, the degrees of anisotropy, the structure in velocity space and the relation between density and velocity structure. Deviations from a general self-similar behaviour provide the clue to estimate the relative influence of different physical processes driving structure formation and thus to understand the turbulent nature of molecular clouds.

We first use a general differential analytical first order approximation to show that, in some realistic circumstancies, the time evolution of the Star Formation Rate in the Solar neighbourhood may behave chaotically in the sense of the deterministic chaos of the well known logistic map. We then use the “phase space reconstruction” technique to demonstrate the existence of chaotic behaviour in the observational data: both in the observed number of stars formed as a function of age, and in the metallicity of stars as a function of age. We conclude by deducing a positive Lyapounov exponent for the data of number of stars vs. age as a clear mark of chaos.

A recent re-analysis of the Egret data by Dixon et al. has shown the existence of a statistically significant diffuse γ-ray emission from the galactic halo. This emission is naturally explained within a previously-proposed model for baryonic dark matter, in which γ-rays are produced through the interaction of high-energy cosmic-ray protons with cold gas clouds clumped along with brown dwarfs into dark clusters. These dark clusters supposedly populate the outer galactic halo and can show up in microlensing observations.

Ion-molecule radiative association reactions are considered to be of relevant importance in the interstellar medium. First results on radiative association rate coefficients were obtained with ion cyclotron resonance (ICR) technique. In an ICR apparatus the pressure can be varied over a wide range, and direct information on the radiative and ternary association reactions can be obtained. Most of the laboratory studies have been concerned primarily with collisional stabilisation of the intermediate (AB⁺)* complex whose important parameter is the lifetime with respect to unimolecular dissociation. We report here the results of ion-molecule reactions between polycyclic aromatic hydrocarbon (PAHs) cations and H₂0, NH₃, H₂, CO studied by the ICR technique. Previously these reactions were studied by the SIFT technique at 0.5Torr and the reaction of dehydrogenated naphthalene cations with hydrogen molecules was also studied with ICR. The radical! catio ns of PAHs are unreactive with H₂, CO, H₂O and NH₃, whilst dehydrogenated PAHs readily associates with H₂ and other molecules. These studies suggest that the protonated ions are likely to be the dominant forms of PAH cations in the interstellar medium in environments where the parent radical cation is itself present.

The emission from a charged particle moving in a medium containing randomly distributed dust particles is considered. It is shown that in certain cases, diffusional mechanism of emission is the principal one in the x-ray range. Possible astrophysical applications of this mechanism are considered. Spectral indices in the x-ray range are calculated, in particular. The calculated spectral indices agree well with those of active galactic nuclei.

I overview the GRAPE project to develop special-purpose computers for astrophysical N-body problems. First I briefly discuss the basic idea behind GRAPE and describe the project history, and then present new hardwares, GRAPE-5 and GRAPE-6, and some important new scientific results. GRAPE-5 is the successor of GRAPE-3, which has been installed in more than 30 institutes all over the world. GRAPE-5 achieved a factor-of-ten improvement in calculation speed, communication speed and accuracy over GRAPE-3. GRAPE-6 is the successor of the Teraflops GRAPE-4, and the planned peak speed of GRAPE-6 exceeds 100 Tflops. For scientific results, I’ll focus on the structure of the central region of galaxies with and without massive black holes.

This paper explains the implications of a mathematical theory (by the same authors) of holes in fractals and their relation to dimension for measurements. The novelty of our approach is to consider the fractal measure on a set rather than just the support of that measure. This should take into account in a more precise way the distribution of data points in measured sets, such as the distribution of galaxies.

The treatment of long-range interacting systems (including Newtonian selfgravitating systems) remains a challenging issue in statistical mechanics. Due to the lack of extensivity, they present non-standard effects like negative specific heats, which implies the inequivalence of statistical ensembles (namely, microcanonical and canonical) even in the limit of infinite number of particles (N → ∞ ). In this paper we review a series of results obtained for one and two dimensional simple N-body dynamical models with infinite-range attractive interactions and without short distance singularities. The free energy of both models can be exactly obtained in the canonical ensemble, while information on the microcanonical ensemble and on the dynamical evolution can be derived from direct numerical simulations with simple O(N) codes, which make use of mean-field variables. Both models show a phase transition from a low energy clustered phase to a high energy gaseous state, in analogy with the models introduced in the early 70’s by Thirring and Hertel.The phase transition is second order for the 1D model, first order for the 2D model. Negative specific heat appears in both models near the phase transition point, but while for the 2D model it is an equilibrium phenomenon, in the 1D case it is typical of transient metastable states, whose lifetime grows with N. For both models, in the presence of a negative specific heat, a cluster of collapsed particles coexists with a halo of higher energy particles which perform long correlated flights, which lead to anomalous diffusion: the mean square displacement grows faster than linear with time.

We explore the behaviour of the solutions of the eqs. of motion derived from the minisuperspace Hamiltonian of the Bianchi IX cosmological model, giving a gaugeinvariant description of the full continuous dynamics and of its stability properties. The study of the latter is performed through a geometrical description of dynamics in the framework of the the theory of Finsler Spaces.

Cosmological model describing the evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electro-magnetic composite pbrane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N -1)-dimensional Lobachevsky space H^N−1, N = n +l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of a sphere S^N−2 by point-like sources. Some examples with billiards of finite volume and hence oscillating behaviour near the singularity are considered. Among them examples with square and triangle 2-dimensional billiards (e.g. that of the Bianchi-IX model) and a 4-dimensional billiard in “truncated” D = 11 supergravity model are considered.

We derive an asymptotic solution of the Einstein field equations which describes the propagation of a thin, large amplitude gravitational wave into a curved spacetime. The resulting equations have the same form as the colliding plane wave equations without one of the usual constraint equations.

The problem of the apparent inconsistency between the standard theory of twobody relaxation and the results about the exponential instability and related stochasticity of N-body systems is here addressed. The solution comes from a careful discussion of the treatment of the phase-space evolution, showing how the distinction between the microscopic instability of the trajectories and the diffusion in action space is able to produce a formally consistent picture.

On the framework of a top-down large scale structure formation, we show a property of scale invariance of the successive fragmentation process. This is a consequence of the microscopical properties of the matter that drives the cosmology. Our results of are independent on the mass of dark matter fermion particles and on the red shift z at which the fragmentation starts.

The two point correlation function (CF) of galaxies is used in the study of large scale structure of the universe. A double power law, with a cut-off radius for the CF of the galaxies and of the clusters has been recovered. A simple fixed point equation for cut-off radius, obtained under very fair physical assumptions, explains this cut-off radius in the framework of the Cellular Structure of the Universe.

The temporal stucture of chaos in three-body dynamics is analysed and described in terms that enable one to make a comparison with standard patterns of chaotic behaviour known in nonlinear physics. With the use of time series generated by three-body systems in computer simulations, we demonstrate intermittent behaviour of the systems with quasi-regular and chaotic states alternating in an unpredictable manner. A fractal dimension of the time series is found to be between 2 and 2.1. An analog of strange attractor with this dimension is assumed behind the time series. Kolmogorov-Sinai entropy is estimated for three-body dynamics and its relation to the Lyapunov exponent is discussed.

The conditions of syzygy crossing leading to slingshot-escape are investigated in the planar three-body problem with equal masses. By systematic numerical surveys of escape orbits with zero-initial velocities, it is found that velocity vectors of the temporary binary are spread out when the third particle passes through between them. The critical value of the velocity-vector products is also found. This condition is not equivalent to the positive radial-velocity of the binary.

The power spectra of time series, which describe different processes in the Sun, solar atmosphere, solar wind and Earth magnetosphere are analyzed within the time range from few days to few years. In accordance with the developed approach, spectral indices are regarded as dynamic parameters, which characterize the time of relaxation of fluctuations in the examined processes.

Thermodynamic stability in relativistic regime is analysed for spherically symmetric selfgravitating systems. The equilibrium configurations are developed by using a distribution function with a cutoff in the energy as a function of Schwarzschild metrics. Stability analysis was considered in several cases in Newtonian regime where the results are known in literature. The situation is different in relativistic regime where the results are still unclear. In this work the problem is clarified by obtaining the critical curve of the onset of instability in the plane z_c-T (central redshift-temperature) of the equilibrium configurations. This curve under particular conditions coincides with the one connected to dynamical instability while, for cooled systems, separates. The curve of the onset of dynamical instability never reaches Newtonian regime while the thermodynamic one reaches it, giving a critical value of the central gravitational potential in complete accordance to the classical results known in Newtonian regime.

The chaotic motion near separatrices of orbital resonances in planetary satellite systems is considered. A separatrix map is constructed, describing the motion in a vicinity of the separatrix of a model orbital resonance. The derived separatrix map is algorithmic: it contains conditional transfer statements. Besides, it is biresonant, i.e. the phase space of the map contains two distinct primary resonance cells. The map is applied to studies of a particular problem of orbital dynamics of a system of two Uranian satellites, Miranda and Umbriel. Phase portraits of the map perfectly reproduce surfaces of section of the phase space of an orbital resonance which this system may have encountered during its long-time evolution. Analysis of the map allows one to straightforwardly investigate the structure of the near-separatrix phase space.

Two classes of lemon-shaped billiards are investigated. Classical dynamics of these systems exhibits the mixed behavior. Level density fluctuations in the quantal spectrum are analyzed. The validity of the Bohigas-Giannoni-Schmit conjecture for these systems is tested by comparing classical and quantal measures of chaos. Another typical Hamiltonian system, essentially a gravitationally driven Fermi oscillator, is presented and discussed as a useful didactic example for chaotic behavior.

The problem of finding an appropriate geometrical/physical index for measuring a degree of inhomogeneity for a given space-time manifold is posed. Interrelations with the problem of understanding the gravitational/informational entropy are pointed out. An approach based on the notion of approximate symmetry is proposed. A number of related results on definitions of approximate symmetries known from literature are briefly reviewed with emphasis on their geometrical/physical content. A definition of a Killing-like symmetry is given and a classification theorem for all possible averaged space-times acquiring Killing-like symmetries upon averaging out a space-time with a homothetic Killing symmetry is proved.

We introduce a set of phenomenological relations connecting characteristic quantities of large scale cosmological structures to the fundamental quantum scales of nucleons. The relations lead to predictions on the relevant mechanical and thermodynamical properties of galaxies, which are in excellent agreement with the experimentally observed data.

We consider the effect of a random magnetic field in the convective zone of the Sun on resonant neutrino spin-flavour oscillations. We argue for the existence of a field of strongly chaotic nature at the bottom of the convective zone. The expected signals in the different experiments (SK,GALLEX-SAGE,Homestake) are obtained as a function of the level of noise, regular magnetic field and neutrino mixing parameters. Previous results obtained for small mixing and ad-hoc regular magnetic profiles are reobtained. We find that MSW regions are stable up to very large levels of noise (P=0.7-0.8) and they are acceptable from the point of view of antineutrino production. For strong noise any parameter region (Δm²,sin²2θ) is excluded: this model of noisy magnetic field is not compatible with particle physics solutions to the SNP. One is allowed then to reverse the problem and to put limits on r.m.s field strength, correlation length and transition magnetic moments by demanding a solution to the SNP under this scenario.

The following sections are included

• Introduction

• Einstein Democracy: the cosmological principle applied to all spacetime events

• Friedmann-Gamow-Fermi Democracy: the cosmological principle applied to all space points

• The New Democracy: the cosmological principle applied to all observers

• References

The following sections are included:

• AUTHOR INDEX